1 Introduction

Nanofluid can be offered as an applicable way to improve heat transfer. Nanofluid convective simulation has been investigated by different researchers^[1-5]. Sheikholeslami and Sadoughi^[6] reported nanofluid convective flow in existence of melting surface. Sheikholeslami and Rokni^[7] published a review article about various applications of magnetic nanofluid. A comprehensive review paper was published by Sheikholeslami and Ganji^[8] to show importance of nanotechnology. The influence of radiation mode was examined by Hayat et al.^[9]. The effect of Coulomb forces on nanofluid behavior was demonstrated by Sheikholeslami and Chamkha^[10]. Their outputs revealed that the electric field is highly sensible in lower Reynolds numbers. Hassan et al.^[11] showed an innovative model for predicting solar radiation. Nayak et al.^[12] reported the roles of nanofluid radiative heat transfer.

The effect of shape factor on nanofluid properties was considered by Sheikholeslami and Bhatti^[13]. Tao and He^[14] presented free convection of nanofluid in an energy storage system. Makinde et al.^[15] demonstrated the nanofluid flow considering non-uniform viscosity. Mezrhab et al.^[16] reported the radiation impact in an enclosure. Sheremet et al.^[17] illustrated the transient ferrofluid flow in a cavity by means of the finite difference method. Some researchers also used nanofluid as effective working fluid^[18-41].

This research aims to model the effects of thermal radiation on nanofluid behavior in existence of Coulomb forces. The roles of Darcy number, radiation parameter, supplied voltage, volume fraction of nanofluid, and Reynolds number are demonstrated in results.

2 Problem statement

The ethylene glycol-Fe₃O₄ nanofluid is utilized. All walls are stationary except for the bottom wall. Figure 1 demonstrates a sample element and geometry. The influence of Re and Da on contour of q is demonstrated in Fig. 2. The effect of Re on q is less sensible than Da. As the Darcy number augments, the shape of isoelectric density lines becomes more complex.

3 Governing formula and modeling 3.1 Governing formula

The definition of electric field is^[23]

(1)

(2)

(3)

(4)

The governing formulae are^[23]

(5)

(ρC_p)_nf, μ_nf, and ρ_nf are^[26]

(6)

Properties of Fe₃O₄ and C₂H₆O₂ are illustrated in Table 1^[26]. Table 2 shows the coefficient values of A_i (i=1, 2, 3, 4)^[7]. k_nf can be obtained from

(7)

Table 3 depicts various shape factors.

Therefore, the final partial differential equations are

(8)

where

(9)

Ψ and Ω are employed in order to diminish the pressure gradient,

(10)

The local Nusselt number Nu_locand the average Nusselt number Nu_aveover the bottom wall are

(11)

(12)

3.2 CVFEM

In order to estimate scalars, we utilize linear interpolation in the triangular element (see Fig. 1(b)). A Gauss-Seidel tool is employed to obtain the final answer after discretization^[29].

4 Mesh study and code validation

Various mesh sizes are tested to find the independent result of the mesh. Table 4 demonstrates an example. This table indicates that the size of 81× 241 can be selected. The CVFEM code is validated by comparing the results with those published in Refs.[23] and [25] (see Fig. 3). Good agreement can be found.

5 Results and discussion

Electrohydrodynamic nanofluid forced convection in presence of thermal radiation is reported. The porous enclosure is filled with Fe₃O₄-ethylene glycol and has one lid wall. Roles of the Darcy number (Da=10² to 10⁵), the supplied voltage (Δφ =0kV to 10kV), the volume fraction of Fe₃O₄ (φ =0% to 5%), the radiation parameter (R_d=0 to 0.8), and the Reynolds number (Re=3000 to 6000) are illustrated graphically.

At first, the impact of the shape factor on the rate of heat transfer is reported in Table 5. In this table, various shapes of nanoparticles are utilized. The maximum Nu happens by platelet. Therefore, platelet nanoparticles are utilized for more investigation.

Figures 4-7 depict the impacts of Da, Re, and Δφ on isotherms and streamlines. At low Re, there is one clockwise vortex in streamline. The midpoint of main vortex is near the positive electrode. Augmenting the Darcy number leads to generation of the second eddy which rotates counter clockwise and the center of main eddy shift to upper side. Applying the electric field causes the strength of the main vortex to enhance and shift the midpoint of the eddy to upper side. Isotherms become more disturbed when Δ φ ≠ 0kV. Thermal plume appears by increasing the Reynolds number. Also, by augmenting Re, ψ_max augments. As the Coulomb force increases, the secondary eddy diminishes and the strength of main eddy enhances.

Nu_aveversus Re, Da, R_d, and Δφ is depicted in Fig. 8. The related formula is

(13)

where Re^* =0.001Re. In absence of the Coulomb force, the Nusselt number augments with the increase of Reynolds number, while an opposite behavior is reported in existence of such forces. The electric field helps the convective mode enhance. Therefore, Nu_aveaugments with the increase of Δφ. Thermal radiation enhances the temperature gradient near the lid wall. The influence of Darcy number is the same as the radiation parameter. Therefore, Nu_aveis an increasing function of R_d and Da.

6 Conclusions

Forced convection and radiation of nanofluid inside a lid driven permeable media in existence of electric field are modeled. Outputs are reported for different values of Da, R_d, φ, Δφ, and Re. Outputs demonstrate that the shape of isotherms becomes more complex with the augment of Da, R_d, and Coulomb forces. Applying the electric field makes the secondary eddy to diminish. The temperature gradient enhances with the increase in the radiation parameter.